Measurement of a set

Measure of a set . Denote a measure of an interval (a, b) , where a <b is its length ba.

Let H be an open set bounded by the line. There is a proposition that H is equal to the countable union of open intervals two to two disjoint, any interval is (a j , b j ) .It is called the measure m (H) of the bounded open set H the sum of the lengths of its intervals (b j – a j .

It should be noted that if the number of the intervals (a j , b j ) is countable, then the sum of the lengths of the intervals is a numerical series, with positive terms (b j – a j ). Because H is a bounded set, this series is convergent.

Case of an arbitrary set

Let F be an arbitrary bounded set. All sorts of open sets H containing F are analyzed. The set {mH} of the measures of these sets is bounded inferiorly, ie 0, and therefore has the smallest, inf {mH}.

The number m * F = inf {mH} is called the outer measure of the set F.

 

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